Formally, the ring R is leftNoetherian iff one (and therefore all) of the following equivalent conditions hold:
Every field is trivially Noetherian, since a field F has only two ideals  F and {0}. Every finite ring is Noetherian. Other familar examples of Noetherian rings are the ring of integers, Z; and Z[x], the ring of polynomials over the integers. In fact, the Hilbert basis theorem states that if a ring R is Noetherian, then the polynomial ring R[x] is Noetherian as well. If R is a Noetherian ring and I is an ideal, then the quotient ring R/I is also Noetherian. Every commutative Artinian ring[?] is Noetherian.
An example of a ring that's not Noetherian is a ring of polynomials in infinitely many variables: the ideal generated by these variables cannot be finitely generated.
Noetherian rings are named after the mathematician Emmy Noether, who developed much of their theory.
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